#453907

order	512
nilpotency class	3
isoclinism	isoclinic class of #428059
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

587 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(9,10,13,19,15,20,14,17)(11,21,22,31,25,33,24,16)(12,26,28,18,30,34,29,36)(23,39,27,37,40,32,38,35);(1,2,4,6)(3,7,8,5)(9,11)(10,16)(12,27)(13,24)(14,22)(15,25)(17,21)(18,35)(19,33)(20,31)(23,28)(26,39)(29,40)(30,38)(32,34)(36,37);(1,3,4,8)(2,5,6,7)(10,17,20,19)(11,22,25,24)(16,31)(18,34,36,26)(21,33)(23,38,40,27)(32,39)(35,37);(9,12,14,28,15,30,13,29)(10,18,19,26,20,36,17,34)(11,23,22,38,25,40,24,27)(16,32,33,35,31,39,21,37)

Polycyclic presentation:

2^9|a^2=e,b^2=gh,c^2=fgi,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361103);